Use of circular cylinders as surrogates for
hexagonal pristine ice crystals in scattering
calculations at infrared wavelengths
Yong-Keun Lee, Ping Yang, Michael I. Mishchenko, Bryan A. Baum, Yong X. Hu,
Hung-Lung Huang, Warren J. Wiscombe, and Anthony J. Baran
We investigate the errors associated with the use of circular cylinders as surrogates for hexagonal
columns in computing the optical properties of pristine ice crystals at infrared 共8 –12-␮m兲 wavelengths.
The equivalent circular cylinders are specified in terms of volume 共V兲, projected area 共 A兲, and volumeto-area ratio that are equal to those of the hexagonal columns. We use the T-matrix method to compute
the optical properties of the equivalent circular cylinders. We apply the finite-difference time-domain
method to compute the optical properties of hexagonal ice columns smaller than 40 ␮m. For hexagonal
columns larger than 40 ␮m we employ an improved geometric optics method and a stretched scattering
potential technique developed in previous studies to calculate the phase function and the extinction 共or
absorption兲 efficiency, respectively. The differences between the results for circular cylinders and hexagonal columns are of the order of a few percent. Thus it is quite reasonable to use a circular cylinder
geometry as a surrogate for pristine hexagonal ice columns for scattering calculations at infrared
共8 –12-␮m兲 wavelengths. Although the pristine ice crystals can be approximated as circular cylinders in
scattering calculations at infrared wavelengths, it is shown that optical properties of individual aggregates cannot be well approximated by those of individual finite columns or cylinders. © 2003 Optical
Society of America
OCIS codes: 010.1290, 010.1310, 010.3920, 290.5850, 290.1090, 280.1310.
1. Introduction
It is quite challenging to assess quantitatively the
radiative effect of cirrus clouds in the atmosphere.1–3
One of the major difficulties encountered in modeling
the radiative properties of cirrus clouds is that these
clouds are composed almost exclusively of nonspherical ice crystals. Although various ice crystal config-
Y.-K. Lee and P. Yang 共pyang@ariel.met.tamu.edu兲 are with the
Department of Atmospheric Sciences, Texas A&M University, College Station, Texas 77843. M. I. Mishchenko is with the NASA
Goddard Institute for Space Studies, 2880 Broadway, New York,
New York 10025. B. A. Baum and Y. X. Hu are with the NASA
Langley Research Center, MS 420, Hampton, Virginia 23681.
H.-L. Huang is with the Cooperative Institute for Meteorological
Satellite Studies, University of Wisconsin—Madison, 1225 West
Dayton Street, Madison, Wisconsin 53706. W. J. Wiscombe is
with NASA Goddard Space Flight Center, Code 913, Greenbelt,
Maryland 2077. A. J. Baran is with the Met Office, Bracknell,
Berkshire, RG12 2 SZ, UK.
Received 24 September 2002; revised manuscript received 6 January 2003.
0003-6935兾03兾152653-12$15.00兾0
© 2003 Optical Society of America
urations, including hexagonal columns, plates,
hollow columns, bullet rosettes, and aggregates, have
been observed,4 –7 it is a common practice to assume
that the geometry of ice crystals is that of hexagonal
columns in climate and remote-sensing applications.8 –11 This simplification is based mainly on two
facts: First, the halos observed at visible wavelengths are scattering characteristics that are typical
of a hexagonal crystal geometry, although it has been
argued that halos are not often observed in the atmosphere 共see, e.g., Foot12兲. Second, pristine ice
crystals that form at extremely cold temperatures,
such as in cirrus clouds and polar stratospheric
clouds, tend to have basic hexagonal structures.
Various parameterization efforts and radiative sensitivity studies assume a hexagonal geometry for ice
crystals in cirrus clouds. However, in a few recent
studies a mixture of various ice crystal configurations
was assumed,13–18 but those studies were essentially
limited to the solar spectrum.
At solar wavelengths, particularly at visible wavelengths, the scattering properties of hexagonal ice
crystals can be approximated by use of the geometric
optics ray-tracing method. At infrared wave20 May 2003 兾 Vol. 42, No. 15 兾 APPLIED OPTICS
2653
lengths, for which the sizes of ice crystals are not
substantially larger than the incident wavelength,
the geometric optics method breaks down. In addition, ice crystals have strong absorption at infrared
wavelengths. Because of the substantial absorption
within ice crystals at infrared wavelengths, the electromagnetic wavelengths refracted into ice crystals
are inhomogeneous waves.19,20 The inhomogeneity
of the refracted electromagnetic wave can significantly complicate the ray-tracing computation and
degrade the accuracy of the conventional ray-tracing
technique.21 Therefore, accurate scattering computational methods must be used in calculating the
scattering properties of ice crystals at infrared wavelengths. The finite-difference time-domain 共FDTD兲
method22–25 and the discrete dipole approximation26 –28 共DDA兲 are accurate numerical methods that
can be applied to an arbitrarily shaped particle, assuming that a sufficiently fine resolution is used for
the grid meshes 共FDTD兲 or the sizes of the dipole
elements 共DDA兲. However, these two methods require extremely large computational resources in
terms of computer CPU time and memory. The applicability of the FDTD and the DDA is quite limited
in practice, particularly when scattering calculations
are required for various size parameters and multiple
wavelengths. The T-matrix method, if it is implemented for axisymmetric particles, is the most computationally efficient method available now for
accurately computing the single-scattering properties of nonspherical particles.29
In the present study we ascertain the accuracy of
using circular cylinders as surrogates for computation of the scattering properties of hexagonal ice crystals at infrared wavelengths. The hexagonal
geometry is regarded as more complex than the circular cylinder geometry, as the former has a lower
degree of symmetry. The accuracy of using a simpler geometry as a surrogate for a more-complex geometry in light-scattering computations was
investigated in several previous studies. Liou and
Takano30 showed that an equivalent sphere with the
same surface area or the same volume does not reproduce the proper single-scattering properties of
hexagonal ice crystals at infrared wavelengths.
Chylek and Videen31 also showed that the equivalent
spheres of equal volume or equal surface area are not
suitable for approximating hexagonal columns or
plates. Macke and Mishchenko32 investigated the
accuracy of approximating a hexagonal geometry by
using ellipsoidal and circular cylinders in lightscattering calculations at visible and near-infrared
wavelengths. Those authors found substantial differences in light-scattering calculations for three geometries 共hexagonal, ellipsoidal, and circular
cylinder particles兲 at a nonabsorbing wavelength
共e.g., 0.55 ␮m兲 and recommended against the substitution of ellipsoidal and circular cylinder geometries
for the hexagonal structure. However, for the integrated scattering properties such as the asymmetry
factor at absorbing wavelengths 共e.g., 1.6 and 3.7
␮m兲, the overall differences among the three geome2654
APPLIED OPTICS 兾 Vol. 42, No. 15 兾 20 May 2003
tries are much smaller in magnitude. Kahnert et al.33
investigated the accuracy of approximating an ensemble of wavelength-sized prisms by spheroids and
cylinders in light-scattering calculations based on the
extended boundary condition method. Their results
show that the optical properties of cylinders are
closer to those of prisms than of spheroids. Baran et
al.34 investigated the accuracy of using a size–shape
distribution of randomly oriented circular ice cylinders to simulate scattering from a distribution of randomly oriented ice aggregates. Grenfell and
Warren35 suggested that a nonspherical particle can
be approximated by a collection of monodisperse
spheres that have the same volume-to-surface 共V兾A兲
ratio. Those authors also carried out intensive validation studies of the accuracy of their method for
calculating the bulk optical properties of ice crystals
and modeling radiative transfer processes that involve ice clouds. The present study is an extension
of that of Grenfell and Warren35 in the sense that a
simpler geometry is used to approximate a morecomplex geometry for scattering and absorption computations.
Motivated by the computational efficiency of the
T-matrix method when it is applied to a circular cylinder geometry with small and moderate size parameters, one may inquire whether a hexagonal
geometry can be approximated by a circular cylinder
at infrared wavelengths with acceptable errors in the
scattering properties. From physical intuition, one
may expect that the detailed sharp edges of side faces
of a hexagon may not be so important in lightscattering calculations in the infrared region because
of strong absorption within the particle and because
the wavelength is of the same order as the particle
size. In addition, in the atmosphere the surface
edges of pristine ice crystals may be rounded owing to
sublimation or riming.32 Thus an investigation of
the scattering properties of circular cylinders may be
interesting in its own right. The efficient computation of single-scattering properties for circular cylinders by the T-matrix method may provide the optical
properties of small ice crystals and also provide a
data set for constructing the approximate optical
properties of large particles based on the composite
method developed by Fu et al.,36,37 which is one of the
most practical approaches available for covering a
wide range of size parameters.
This paper is organized as follows. In Section 2
we present the approach used for light-scattering calculations. Seven ways to approximate a hexagonal
geometry by a circular cylinder are presented. Numerical computations and an associated discussion
are provided in Section 3. Finally, conclusions are
given in Section 4.
2. Approach
In the present study we use the T-matrix computation program developed by Mishchenko38 to calculate
the single-scattering properties of randomly oriented
circular cylinders. A detailed description of the
T-matrix method and documentation of the compu-
tational program have been provided by Mishchenko
and Travis39 and will not be reiterated here. The
T-matrix method pioneered by Waterman40 can be
applied in principle to any arbitrary geometry. The
application of the T-matrix method based on the extended boundary condition method for rotationally
symmetric shapes can be traced back to the studies of
Wiscombe and Mugnai,41 Barber and Hill,42 and
Mishchenko and Travis.39 Recently the numerical
implementation of this method was extended to geometries other than axisymmetric particles.43– 45
Havemann and Baran46 employed the T-matrix
method to compute the single-scattering properties of
hexagonal ice crystals with size parameters up to 40.
In the numerical computation, the applicable size
parameter region of the T-matrix, if it is implemented
as a nonaxisymmetric geometry, is usually narrower
than for axisymmetric particles. A combination of
the T-matrix method and other numerical lightscattering computational methods, such as the DDA
and the FDTD, may shed new light on efficient computation of the optical properties of nonspherical particles.47,48 In practice, the analytical approach of
averaging the effect of particle orientations in the
T-matrix method49 can substantially speed up numerical computations.
To ascertain the difference in the optical properties
of hexagonal and circular cylindrical ice crystals we
take the extinction efficiency, the single-scattering
albedo, the phase function, and the asymmetry factor
calculated by Yang et al.50 as the reference data set.
In that study the FDTD method was applied to small
hexagonal ice crystals with maximum dimensions
smaller than 40 ␮m. For the computation of the
asymmetry factor, Yang et al.50 used a combination of
the FDTD method and an improved geometric optics
method 共for particle sizes larger than 40 ␮m兲 to compute the phase function. In the computation of extinction and absorption efficiencies, the stretched
scattering potential method 共SSPM兲 is applied to ice
crystals with maximum dimensions larger than 40
␮m. The SSPM results are refined by the weighted
summation of the SSPM, the Lorenz–Mie solution for
equivalent spheres, and the geometric optics solution
in a manner similar to the composite approach developed by Fu et al.36 When the FDTD and the refined
SSPM solutions are combined, the results computed
by Yang et al.50 encompass ice crystal sizes specified
in terms of their maximum dimensions from 1 to
10000 ␮m.
In the present study we consider particle sizes that
range from 1 to 180 ␮m. Over this size range, the
T-matrix method gives convergent solutions in the
8 –12-␮m spectral region. For a given maximum dimension, the aspect ratio given by Yang et al.50 is
followed, which is
再
where a is the semiwidth of the cross section and L is
the length of a hexagonal column. Ice crystals defined by Eq. 共1兲 are compact hexagons with aspect
ratios of unity if their maximum dimensions are
smaller than 40 ␮m, whereas crystals larger than 40
␮m are essentially hexagonal columns.
The radius of the cross section and the length of a
circular cylinder are denoted R and H, respectively.
To define an equivalent circular cylinder for a hexagonal particle, one can assume that the two particles
have the same projected area, volume, or ratio of
volume to projected area under the condition that the
two particles have the same length or aspect ratio.
If circular and hexagonal cylinders have the same
length 共i.e., H ⫽ L兲, the radius of the circular cylinder
with the same volume is given by
Rv ⫽
冑冑
3 3
a,
2␲
(2)
where the subscript v indicates that the circular cylinder has the same volume as the hexagon. The
cross-sectional radius of a circular cylinder with the
same projected area as the hexagon is given by
Ra ⫽
关L 2 ⫹ 共6 冑3a 2 ⫹ 12aL兲兾␲兴 1兾2 ⫺ L
.
2
(3)
Similarly, the cross-sectional radius of the circular
cylinder that has the same ratio of volume to projected area as a hexagon is given by
R v兾a ⫽
冑3
2
a.
(4)
To define the equivalence of a circular cylinder and
a hexagonal column in scattering calculations, one
can also let the two particles have the same aspect
ratio, that is, a兾L ⫽ R兾H. For this condition the
cross-sectional radius of a circular cylinder with an
equivalent volume, an equivalent projected area, or
an equivalent ratio of volume to projected area is
given by
1
2a兾L ⫽ exp关⫺0.017835共L ⫺ 40兲兴
5.916兾L 1兾2
R v* ⫽
R a* ⫽
R v兾a* ⫽
冉 冊
冋冑 册
3 冑3
2␲
1兾3
a,
3 3a ⫹ 6L
2␲共a ⫹ L兲
(5)
1兾2
a,
冑3共a ⫹ L兲
a.
冑3a ⫹ 2L
L ⱕ 40 ␮m
40 ␮m ⬍ L ⱕ 50 ␮m,
L ⬎ 50 ␮m
20 May 2003 兾 Vol. 42, No. 15 兾 APPLIED OPTICS
(6)
(7)
(1)
2655
Fig. 1. Radii and lengths of circular cylinders defined as having
the same volume, surface area, or ratio of volume to surface area
as hexagonal columns when the same length or aspect ratio is
applied to the two geometries. Left, same lengths of circular cylinders and hexagonal columns; Middle and right, aspect ratio kept
constant in defining the equivalence.
The lengths associated with the radii in Eqs. 共5兲–共7兲
are given by
冉 冊
冋冑 册
3 冑3
H v* ⫽
2␲
1兾3
L,
3 3a ⫹ 6L
H a* ⫽
2␲共a ⫹ L兲
H v兾a* ⫽
(8)
1兾2
L,
冑3共a ⫹ L兲
L.
冑3a ⫹ 2L
(9)
(10)
Figure 1 shows the radii and lengths defined in
Eqs. 共3兲–共7兲 versus semiwidth 共a兲 and length 共L兲 of a
hexagonal column. The aspect ratio defined in Eq.
共1兲 is used for defining the semiwidth of the cross
section of a hexagonal column of a given length. At
the left in Fig. 1 the radius of a circular cylinder with
the same surface 共 A兲, volume 共V兲, or ratio of volume
to surface area 共V兾A兲 as a hexagonal column when
the lengths of the two geometries are the same is
shown. Evidently, the circular cylinder specified on
the basis of surface-area equivalence is largest,
whereas that based on V兾A equivalence is smallest in
terms of the radius of cross section. In the middle
and at the right in Fig. 1 are the radii and lengths of
cylinders with aspect ratios that are the same as for
the hexagonal crystals.
3. Numerical Results and Discussion
The results presented here focus on the scalar optical
properties of scattering particles, including extinction efficiency, absorption efficiency, phase function,
and asymmetry factor. Similarly to Yang et al.,50 we
use the refractive index compiled by Warren51 in our
numerical computations. The T-matrix computational code is implemented with the extended double
precision algorithm.52
Figure 2 shows a comparison of the phase functions
2656
Fig. 2. Comparison of phase functions of circular cylinders and
hexagonal columns. The circular cylinders are defined to have
the same volume and length as the hexagonal columns. The results for circular cylinders are computed by Mishchenko’s T-matrix
code.38 For hexagonal columns the FDTD method is used for
small particles 共L ⫽ 10, 20, 40 ␮m兲, whereas an improved geometric optics method is used for L ⫽ 140 ␮m.
APPLIED OPTICS 兾 Vol. 42, No. 15 兾 20 May 2003
of hexagonal ice columns and circular ice cylinders.
The circular cylinders are defined to have the same
length and volume as the hexagonal columns. For
sizes L ⫽ 10, 20, 40 ␮m, the phase functions of the
hexagonal columns are computed by the FDTD
method, whereas the results for L ⫽ 140 ␮m are
computed with an improved geometric optics method.
For small sizes, the phase functions of circular cylinders are essentially the same as those of hexagonal
columns. The slight differences between the two results near backscattering angles for L ⫽ 40 ␮m might
be caused by the inaccuracy of the FDTD method
because of insufficient resolution for the grid mesh.
The performance of the FDTD method for hexagonal
ice crystals was recently assessed by Baran et al.53 in
comparison with the implementation of the T-matrix
method in a hexagonal geometry. From Fig. 2 excellent agreement between the results at L ⫽ 140 ␮m
can be noted. Evidently, the sharp edges of side
faces of hexagonal geometry are not important in
specifying the optical properties of the particles. Instead, the overall morphology of the particle as a
cylinder or a column is the major factor that affects
the particle’s optical properties.
For infrared radiative transfer simulations, the
phase function of cirrus particles can be approximated by the Henyey–Greenstein 共H-G兲 function,
given by
P H-G共␪兲 ⫽
1 ⫺ g2
共1 ⫹ g 2 ⫺ 2g cos ␪兲 3兾2
N
⫽
兺 共2l ⫹ 1兲 g P 共cos ␪兲,
l
l
l⫽0
(11)
Fig. 3. Comparison of extinction efficiencies at a wavelength of 8.5 ␮m for hexagonal columns and various equivalent circular cylinders.
The results for hexagonal particles are taken from Yang et al.50
where ␪ is the scattering angle and g is the asymmetry factor of ice crystal, which is defined as
g ⫽ 1⁄ 2
兰
␲
P共␪兲cos共␪兲sin共␪兲d␪,
(12)
0
where P共␪兲 is the phase function of nonspherical ice
crystals. Pl共cos ␪兲 in Eq. 共11兲 is the lth Legendre
polynomial. The Henyey–Greenstein phase function is used frequently for its simplicity and efficiency
in numerical computations. In the following discussion, emphasis will be given to the asymmetry factor
of the phase function instead of to the detailed angular variation of the computed scattering phase function. Note that the present computations are
limited to scalar optical properties. The computation of the full phase matrix elements of the circular
ice cylinders at a number of infrared wavelengths
was recently reported by Xu et al.,54 who used Mishchenko’s T-matrix code for their numerical computation.
The present computations cover the terrestrial infrared window 共8 –12-␮m兲 region, but in this paper
the discussion is limited to numerical results at 8.5and 11-␮m wavelengths. The radiometric measurements at the spectral bands centered at these two
wavelengths are often used to retrieve cloud properties.14 For investigations involving ice particles, the
11-␮m wavelength is unique because it is within the
Christiansen band.55,56 To ascertain the differences
between the optical properties of circular ice cylinders and hexagonal ice columns, we define the relative difference ε as follows:
ε⫽
Resultcirc cyl ⫺ Resulthexag col
⫻ 100%.
Resulthexag col
(13)
Figure 3 shows the extinction efficiencies of hexagonal ice columns and various equivalent circular ice
cylinders at ␭ ⫽ 8.5 ␮m. Also shown are the relative
differences between the results for the hexagonal and
the circular cylinder geometries. At the left, the
hexagons and the circular cylinders have the same
length; at the right, the two geometries have the
same aspect ratio. Evidently, for large particles
共⬎120 ␮m兲 the differences among various equivalent
definitions are reduced in magnitude, particularly
when the aspect ratio is kept constant for the two
geometries. The extinction efficiency of the
equivalent-volume circular cylinder 共L ⫽ H兲 is closer
than other equivalence definitions to that of a hexagonal column. The maximum difference is less than
10%, except when the two geometries have the same
length and radius. For large sizes 共⬎100 ␮m兲 the
maximum difference is reduced to approximately 3%.
In a previous study Baran and Havemann57 noticed
that for large size parameters the single-scattering
properties become asymptotic to their limiting values
20 May 2003 兾 Vol. 42, No. 15 兾 APPLIED OPTICS
2657
Fig. 4. Absorption efficiencies that correspond to the extinction efficiencies shown in Fig. 3.
and as such become independent of crystal shape at
infrared wavelengths.
Figure 4 shows the absorption efficiencies and relative differences that correspond to the results shown
in Fig. 3. Evidently, for large particle sizes 共⬎100
␮m兲 the results for various definitions of equivalent
circular cylinders converge, and the relative differences converge to approximately 2%. The differences between the absorption efficiency of hexagonal
columns and that of the circular cylinders with the
same ratio of volume to projected area are smaller
than the results for other equivalence definitions.
Fu et al.36 ascertained the errors of approximating
hexagonal columns by spheres with various equivalent definitions in the computation of absorption efficiency. They also noticed that the equivalent
sphere based on the same ratio of volume to projected
area leads to the smallest errors. Our conclusion is
consistent with the previous study. For practical
applications of infrared radiative transfer, the most
important process is absorption. From this perspective, hexagonal ice columns may be approximated by
circular cylinders with the same ratio of volume to
projected area. In fact, the ratio of particle volume
to particle projected area is proportional to the mean
path length of the rays inside particle in the framework of anomalous diffraction theory.
Figure 5 shows the asymmetry factor values that
correspond to the extinction and absorption efficiencies shown in Figs. 3 and 4. The definition of the
2658
APPLIED OPTICS 兾 Vol. 42, No. 15 兾 20 May 2003
equivalent circular cylinder for a hexagonal column
seems to have a negligible effect on the value of the
asymmetry factor when the particle size is larger
than approximately 80 ␮m. In this case the differences of the asymmetry factors between the hexagonal and circular cylinder geometries are less than 2%.
For particle sizes smaller than 20 ␮m it is important
how the equivalent circular cylinder is defined.
When L ⫽ H, the asymmetry factor for the
equivalent-volume circular cylinders is closer than
the other equivalent circular cylinder definitions to
that of hexagonal columns. Another property shown
in Fig. 5 is that the asymmetry factor is small for
small particles 共less than 15 ␮m兲, whereas the asymmetry factor reaches its asymptotic value for large
particle sizes. For small particles, the scattering
pattern is close to that of Rayleigh scattering and the
phase function is not strongly asymmetric with respect to scattering angle, leading to a small asymmetry factor. When the particle size is large, any rays
refracted into the particles are essentially absorbed
because of the strong absorption of ice at infrared
wavelengths. The scattered energy is derived primarily from the diffracted energy that is concentrated in the forward direction.
Figures 6, 7, and 8 show the extinction efficiency,
the absorption efficiency, and the asymmetry factor,
respectively, but at ␭ ⫽ 11 ␮m. The Christiansen
band lies near 11 ␮m.55,56 In this region the extinction reaches its minimum and the absorption within
Fig. 5. Asymmetry factors that correspond to the efficiencies shown in Figs. 3 and 4.
Fig. 6. Same as Fig. 3, except that the calculations are performed at a wavelength of 11 ␮m.
20 May 2003 兾 Vol. 42, No. 15 兾 APPLIED OPTICS
2659
Fig. 7. Same as Fig. 4, except that the calculations are performed at a wavelength of 11 ␮m.
Fig. 8. Same as Fig. 5, except that the calculations are performed at a wavelength of 11 ␮m.
2660
APPLIED OPTICS 兾 Vol. 42, No. 15 兾 20 May 2003
Table 1. Minimum and Maximum Relative Errors of the Approximation of Hexagonal Column by a Circular Cylinder by Use of a T Matrix Compared
with Reference Results Given by Yang et al.a
H⫽L
a兾L ⫽ R兾H
Equivalent
Surface Area
共A兲
Equivalent
Volume
共V兲
Equivalent
Ratio of V
to A
Equivalent
Surface Area
共A兲
Equivalent
Volume
共V兲
Equivalent
Ratio of V
to A
a ⫽ R and
H⫽L
␭ ⫽ 8.5 ␮m
Extinction efficiency
Absorption efficiency
Asymmetry factor
共2.4, 6.6兲
共2.7, 5.2兲
共0.1, 2.9兲
共⫺0.6, 5.5兲
共2.1, 4.7兲
共⫺0.1, 1.2兲
共⫺6.5, 8.9兲
共⫺1.9, 3.8兲
共⫺6.9, 1.0兲
共0.7, 8.4兲
共2.5, 7.1兲
共⫺0.1, 6.2兲
共1.7, 7.8兲
共2.5, 5.7兲
共⫺0.2, 3.6兲
共⫺0.7, 7.0兲
共2.0, 4.6兲
共⫺3.0, 1.2兲
共⫺2.0, 15.5兲
共2.5, 10.8兲
共⫺0.4, 13.4兲
␭ ⫽ 11 ␮m
Extinction efficiency
Absorption efficiency
Asymmetry factor
共3.8, 5.6兲
共2.7, 6.9兲
共0.0, 3.1兲
共2.3, 5.0兲
共2.3, 6.8兲
共⫺1.8, 0.2兲
共⫺1.2, 4.7兲
共⫺1.1, 6.9兲
共⫺6.9, 0.1兲
共3.9, 6.2兲
共2.1, 7.0兲
共⫺0.1, 5.7兲
共3.9, 5.7兲
共2.2, 6.8兲
共0.1, 3.2兲
共2.2, 4.8兲
共2.3, 6.5兲
共⫺3.6, 0.2兲
共3.9, 9.5兲
共2.0, 9.1兲
共0.1, 12.8兲
Property
a
Ref. 50.
the ice crystal is substantial. Unlike at ␭ ⫽ 8.5 ␮m,
the extinction efficiency at ␭ ⫽ 11 ␮m converges for
various definitions of equivalent circular cylinders for
particle sizes larger than 60 ␮m. The asymptotic
value for the differences between the extinction efficiencies of the hexagonal column and the circular
cylinder is approximately 4%.
For the absorption efficiency shown in Fig. 7, the
results for the equivalent volume, the equivalent surface area, and the equivalent ratio of volume to surface area yield similar differences. All three
definitions yield a maximum relative difference in the
size range 15– 40 ␮m. However, the magnitude of
the differences is less than 7%. For very small ice
crystals, with sizes from 1 to 10 ␮m, the equivalence
based on equal radius and height can lead to differences much larger than differences for other definitions. For the asymmetry factor shown in Fig. 8,
convergence of the results for various equivalence
definitions is obtained for particle sizes larger than
20 ␮m. For small sizes 共⬍20 ␮m兲, the equivalentvolume circular cylinder yields the minimum difference when the two geometries have the same length
共i.e., L ⫽ H兲.
Table 1 lists the errors obtained from approximating hexagonal columns by circular cylinders based on
the calculation of the various optical properties
shown in Figs. 3– 8. Based on these error values, the
equivalence based on the ratio V兾A performs the best
for calculation of the absorption efficiency. For the
asymmetry factor the volume equivalence definition
yields the minimum errors. For extinction efficiency, various equivalence definitions have a similar
error range, except when the equivalence is based on
a ⫽ R and H ⫽ L.
In reality, pristine ice crystals are not common in
cirrus clouds. For ice crystals with complex geometries, their optical properties cannot be well approximated by those of a simple morphological structure.
To illustrate this, in Fig. 9 we show the absorption
efficiencies of pristine hexagonal columns and aggregates. The definition of the aggregate geometry is
explained by Yang and Liou.58 The numerical com-
putation of the optical properties of aggregates in Fig.
9 is explained by Yang et al.59 Following Foot,12
Francis et al.,60 Mitchell and Arnott,61 and Fu et al.,36
we define the effective radius as
Re ⫽
3V
,
4A
(14)
where V and A are the volume and the projected area,
respectively, of the nonspherical particles. From
Fig. 9 it is evident that the absorption efficiencies of
the two geometries are quite different, regardless of
whether the sizes of the particles are defined in terms
of maximum dimension or effective size. Note that
Baran et al.62 ascribe the differences in the absorption efficiencies of aggregates and columns to a tunneling effect.
4. Conclusions
The accuracy of approximating hexagonal crystals by
circular cylinders has been investigated in the computation of the scalar optical properties of pristine ice
crystals in the infrared spectral 共8 –12-␮m兲 region.
Various definitions were used to define the equivalence of particles in a circular cylinder with those of a
hexagonal column. The T-matrix computational
program was used to solve for the single-scattering
properties of circular cylinders.
For extinction efficiency, absorption efficiency, and
asymmetry, the differences between the results for
the two geometries are less than 10%. In general,
errors for particles of sizes smaller than 20 ␮m are
more significant than the errors for larger particles.
For particle sizes larger than 40 ␮m, the differences
are essentially of the order of a few percent. At ␭ ⫽
8.5 ␮m, the circular cylinder with an equivalent
volume-to-surface ratio to that of a hexagonal column
yields a smaller difference in the computation of absorption efficiency than do equivalent-volume or
equivalent-surface circular cylinders. This difference is particularly pronounced for particle sizes less
than 40 ␮m. At ␭ ⫽ 11 ␮m, the definition of equivalence for a circular cylinder has a negligible effect on
20 May 2003 兾 Vol. 42, No. 15 兾 APPLIED OPTICS
2661
Fig. 9. Comparison of absorption efficiencies of pristine hexagonal ice columns and aggregate ice crystals. The procedure for computing
the optical properties of aggregates was explained by Yang et al.59
the calculation of absorption efficiency when particle
sizes are larger than 15 ␮m. A detailed comparison
of the numerical computations associated with various equivalence definitions of circular cylinders indicates that the equivalence based on the ratio V兾A is
most suitable for absorption efficiency when the two
geometries have the same aspect ratio 共a兾L ⫽ R兾H兲
and the volume equivalence is suitable for asymmetry factor or phase function calculations when the two
geometries have the same length 共L ⫽ H兲. The error
ranges for equivalence-area, equivalent-volume, and
equivalent V兾A are slightly different in the computation of extinction efficiency. In general, the errors
associated with the use of circular cylinders as surrogates for hexagonal ice crystals in scattering calculations at infrared wavelengths are of the order of a
few percent. Thus it quite reasonable to approximate the geometry of pristine ice crystals as circular
cylinders in the study of infrared radiation.
It has also been shown that it is not proper to
approximate a complex aggregate geometry with a
simplified geometry such as a hexagonal column for
computing optical properties. Because aggregates
and bullet rosettes are common in cirrus clouds, there
is a need to include their particle geometry in modeling of the optical properties of cirrus particles, even
at infrared wavelengths. Future research will explore a surrogate particle that better approximates
the more-complex crystal geometries found in nature.
In particular, it will be worthwhile to investigate
whether it is valid to approximate an aggregate ice
crystal by using a number of individual cylinders
2662
APPLIED OPTICS 兾 Vol. 42, No. 15 兾 20 May 2003
based on the method developed by Grenfell and Warren,35 who suggested an equivalence of a circular cylinder and a monodisperse sphere 共with the same
volume-to-surface ratio兲 system for scattering and radiative transfer computations.
This study was supported by research grants NAG1-02002 and NAG5-11374 from the NASA Radiation
Sciences Program managed by Donald Anderson and
also by U.S. Office of Naval Research Multidisciplinary Research Program of the University Research
Initiative funded through the Navy Indian Ocean
METOC Imaging mission for the Geostationary Imaging Fourier Transform Spectrometer project as
well as the NASA Cloud-Aerosol Lidar and Infrared
Pathfinder Satellite Observations project. A. J. Baran is partially funded by the U.K. Department of
Environment, Food and Rural Affairs under contract
PECD 7兾12兾37.
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